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Maximizing reliability of the capacity vector for multi-source multi-sink stochastic-flow networks subject to an assignment budget

  • * Corresponding author: M. R. Hassan

    * Corresponding author: M. R. Hassan
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  • Many real-world networks such as freight, power and long distance transportation networks are represented as multi-source multi-sink stochastic flow network. The objective is to transmit flow successfully between the source and the sink nodes. The reliability of the capacity vector of the assigned components is used an indicator to find the best flow strategy on the network. The Components Assignment Problem (CAP) deals with searching the optimal components to a given network subject to one or more constraints. The CAP in multi-source multi-sink stochastic flow networks with multiple commodities has not yet been discussed, so our paper investigates this scenario to maximize the reliability of the capacity vector subject to an assignment budget. The mathematical formulation of the problem is defined, and a proposed solution based on genetic algorithms is developed consisting of two steps. The first searches the set of components with the minimum cost and the second searches the flow vector of this set of components with maximum reliability. We apply the solution approach to three commonly used examples from the literature with two sets of available components to demonstrate its strong performance.

    Mathematics Subject Classification: Primary: 90B25; Secondary: 90B15.

    Citation:

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  • Figure 1.  Modified uniform crossover

    Figure 2.  Mutation operation

    Figure 3.  Crossover operation

    Figure 4.  Mutation operation

    Figure 5.  Network with three source and two sink nodes

    Figure 6.  The minimum cost found at each generation

    Figure 7.  Two-source two-sink computer network

    Figure 8.  The minimum cost found at each generation

    Figure 9.  Network with two source and three sink nodes

    Figure 10.  The minimum cost found at each generation

    Table 1.  Component information for the network in Figure 5

    p Capacity Cost
    0 1 2 3 4
    1 0.02 0.04 0.14 0.80 0.00 1
    2 0.04 0.06 0.10 0.15 0.65 3
    3 0.02 0.03 0.05 0.90 0.00 4
    4 0.05 0.08 0.87 0.00 0.00 2
    5 0.01 0.04 0.10 0.85 0.00 3
    6 0.02 0.05 0.15 0.78 0.00 2
    7 0.05 0.10 0.85 0.00 0.00 1
    8 0.04 0.06 0.15 0.75 0.00 4
    9 0.03 0.05 0.12 0.80 0.00 1
    10 0.01 0.04 0.05 0.15 0.75 3
    11 0.03 0.05 0.07 0.85 0.00 1
    12 0.01 0.02 0.07 0.90 0.00 1
     | Show Table
    DownLoad: CSV

    Table 2.  The initial population for the network example in Figure 5

    No. The components$ \boldsymbol{(}\mathcal{B} $) The Flow vector (F) $ {\boldsymbol{Z}}_{\boldsymbol{obj}}\left(\mathcal{B}\right) $ $ \boldsymbol{R}\boldsymbol{(}{\boldsymbol{X}}_{\boldsymbol{F}}\boldsymbol{(}\mathcal{B}\boldsymbol{)} $
    1 2 10 3 12 7 1 8 11 6 9 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0067 0.259087
    2 1 7 4 11 6 8 9 10 2 3 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0005 0.288896
    3 6 12 8 11 1 10 9 3 2 4 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0164 0.236032
    4 12 6 9 7 11 1 3 8 10 2 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.293038
    5 8 5 4 9 7 1 12 6 10 2 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0036 0.269935
    6 10 7 12 9 1 5 6 3 2 4 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.292652
    7 1 9 6 10 5 3 11 4 2 12 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0000 0.312330
    8 2 1 9 8 12 4 3 6 10 11 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 22.0012 0.282898
    9 9 3 12 5 4 1 6 10 2 7 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 21.0002 0.293636
    10 2 8 4 1 7 5 11 9 10 3 1 0 1 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 23.0000 0.314006
     | Show Table
    DownLoad: CSV

    Table 3.  The component information for the network of Figure 7.

    p Capacity
    0 1 2 3 4 5 6 7 8 9
    1 0.001 0.001 0.003 0.004 0.005 0.005 0.006 0.007 0.010 0.015
    2 0.001 0.003 0.003 0.004 0.005 0.007 0.007 0.008 0.009 0.010
    3 0.002 0.002 0.003 0.006 0.007 0.007 0.010 0.012 0.015 0.017
    4 0.001 0.001 0.002 0.003 0.005 0.008 0.010 0.011 0.012 0.015
    5 0.001 0.002 0.009 0.012 0.020 0.040 0.050 0.060 0.806 0.000
    6 0.001 0.002 0.002 0.005 0.010 0.012 0.015 0.017 0.020 0.025
    7 0.001 0.001 0.002 0.005 0.008 0.010 0.012 0.015 0.015 0.017
    8 0.001 0.002 0.005 0.005 0.007 0.008 0.010 0.012 0.015 0.015
    9 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.008 0.009 0.010
    10 0.002 0.003 0.005 0.006 0.007 0.009 0.012 0.015 0.941 0.000
    11 0.002 0.002 0.003 0.005 0.007 0.008 0.010 0.011 0.020 0.030
    12 0.001 0.002 0.003 0.005 0.008 0.009 0.010 0.012 0.015 0.040
    13 0.001 0.001 0.003 0.005 0.005 0.010 0.011 0.017 0.018 0.020
    14 0.001 0.001 0.002 0.002 0.003 0.005 0.007 0.009 0.016 0.021
    15 0.001 0.001 0.002 0.003 0.004 0.005 0.007 0.008 0.009 0.011
    16 0.001 0.002 0.002 0.004 0.005 0.006 0.007 0.009 0.014 0.017
    17 0.001 0.001 0.002 0.002 0.003 0.004 0.005 0.007 0.009 0.011
    18 0.001 0.001 0.002 0.002 0.002 0.003 0.003 0.004 0.005 0.007
    19 0.001 0.001 0.002 0.003 0.005 0.008 0.009 0.011 0.013 0.014
    20 0.002 0.002 0.003 0.006 0.007 0.007 0.010 0.013 0.015 0.020
    10 11 12 13 14 15 16 17 18 19 20
    0.060 0.150 0.733 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.943 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.919 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.015 0.016 0.020 0.856 0.025 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.891 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.020 0.022 0.025 0.030 0.817 0.000 0.000 0.000 0.000 0.000 0.000
    0.016 0.020 0.884 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.011 0.015 0.016 0.017 0.019 0.020 0.857 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.902 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.895 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.025 0.031 0.853 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.024 0.025 0.030 0.035 0.040 0.060 0.719 0.000 0.000 0.000 0.000
    0.015 0.017 0.020 0.027 0.870 0.000 0.000 0.000 0.000 0.000 0.000
    0.020 0.022 0.025 0.030 0.035 0.040 0.761 0.000 0.000 0.000 0.000
    0.015 0.017 0.018 0.019 0.020 0.022 0.844 0.017 0.017 0.000 0.000
    0.008 0.009 0.011 0.013 0.014 0.014 0.015 0.000 0.000 0.019 0.020
    0.015 0.017 0.020 0.030 0.851 0.000 0.000 0.000 0.000 0.000 0.000
    0.915 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    21 22 23 24 25 26 27 28 29 30 31
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.019 0.020 0.023 0.025 0.026 0.740 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
    0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
     | Show Table
    DownLoad: CSV
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